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This book collects the notes of the lectures given at an Advanced Course on Dynamical Systems at the Centre de Recerca Matemàtica (CRM) in Barcelona. The notes consist of four series of lectures.

The first one, given by Andrew Toms, presents the basic properties of the Cuntz semigroup and its role in the classification program of simple, nuclear, separable C*-algebras. The second series of lectures, delivered by N. Christopher Phillips, serves as an introduction to group actions on C*-algebras and their crossed products, with emphasis on the simple case and when the crossed products are classifiable. The third one, given by David Kerr, treats various developments related to measure-theoretic and topological aspects of crossed products, focusing on internal and external approximation concepts, both for groups and C*-algebras. Finally, the last series of lectures, delivered by Thierry Giordano, is devoted to the theory of topological orbit equivalence, with particular attention to the classification of minimal actions by finitely generated abelian groups on the Cantor set.



The Cuntz Semigroup and the Classification of C*-Algebras


Chapter 1. Introduction

This part of the Advanced Course is meant to introduce the reader to an invariant for C*-algebras —the Cuntz semigroup— which has recently come to the forefront of research on C*-algebra classification. In order to motivate its introduction we return to the roots of the theory of operator algebras, and to the type classification of factors in particular.
Thierry Giordano, David Kerr, N. Christopher Phillips, Andrew Toms

Chapter 2. The Cuntz Semigroup

Here, we introduce the basic theory of the Cuntz semigroup. Any proofs not contained in these notes can be found in Ara–Brown–Guido–Lledo–Perera–Toms [1], but no originality is claimed there. Most of the results here are due to Cuntz [9], Kirchberg–Rørdam [17], and Rørdam [26]. We assume that all C*-algebras are separable unless otherwise noted.
Thierry Giordano, David Kerr, N. Christopher Phillips, Andrew Toms

Chapter 3. Structure of the Cuntz Semigroup

What does the Cuntz semigroup really look like? We will see later that the question is out of reach in any reasonable sense without imposing some conditions on your C*-algebra. For this section, we restrict our attention to unital simple separable C*-algebras which have strict comparison of positive elements. In light of Example 2.0.5, we will also assume that our algebra is stably finite, since the Cuntz semigroup is otherwise degenerate. This means that our algebra admits at least one normalized 2-quasitrace (which will be a trace if the algebra is exact). The material covered in this section can be found in Brown–Perera–Toms [4] and Brown–Toms [5].
Thierry Giordano, David Kerr, N. Christopher Phillips, Andrew Toms

Chapter 4. Elliott’s Program

The Elliott invariant of a C*-algebra A is the 4-tuple
$$\mathrm{Ell}(\mathit{A})\,\, : = \,\, \left(\left(\mathrm{K}_0\mathit{A}, \mathrm{K}_0\mathit{A}^{+}, \sum{}_{\mathit{A}}\right), \mathrm{K}_1\mathit{A}, \,\mathrm{T}^{+}\,(\mathit{A}), \rho\mathit{A} \right),$$
where the K-groups are the Banach algebra ones, K0A+ is the image of the Murray–von-Neumann semigroup V(A) under the Grothendieck map, \(\sum{}_\mathit{A}\) is the subset of K0A corresponding to projections in A, T+(A) is the space of positive tracial linear functionals on A, and ρA is the natural pairing of T+(A) and K0A given by evaluating a trace at a K0-class. The reader is referred to Rørdam’s monograph [27] for a detailed treatment of this invariant. In the case of a unital C*-algebra, the invariant becomes \(\left(\left(\mathrm{K}_0{\mathit{A}}, \mathrm{K}_0{\mathit{A}^{+}}, [{1}_{\mathit{A}}]\right), {\mathrm{K}_1}\mathit{A}, \, \mathrm{T}\,(\mathit{A}), \rho\mathit{A}\right), \), where [1A] is the K0-class of the unit, and T(A) is the (compact convex) space of tracial states. We will concentrate on unital C*-algebras in the sequel in order to limit technicalities.
Thierry Giordano, David Kerr, N. Christopher Phillips, Andrew Toms

Chapter 5. A New Conjecture

The appearance of counterexamples to (EC) in the first half of the 2000s forced a re-evaluation of the classification program Elliott–Toms [14]. The fact that so many natural classes of C*-algebras do satisfy (EC) made it impossible to believe that instances of its confirmation were somehow random. There had to be an underlying principle governing the existing theory and clarifying the line between those algebras which do and do not satisfy (EC). In this section we will introduce and discuss a conjectural version of such a principle, one that has significant evidence to support it. The conjecture relates three regularity properties for C*-algebras which we introduce presently.
Thierry Giordano, David Kerr, N. Christopher Phillips, Andrew Toms

Chapter 6. Nuts and Bolts: Proof Sketches

In this section we take some of the salient results of the preceding section and show more or less how they are proved. We shall take for granted, however, Winter’s locally finite nuclear dimension theorem; a full proof is well beyond the scope of these notes.
Thierry Giordano, David Kerr, N. Christopher Phillips, Andrew Toms

An Introduction to Crossed Product C*-Algebras and Minimal Dynamics


Chapter 7. Introduction and Motivation

These notes are an introduction to group actions on C*-algebras and their crossed products, primarily by discrete groups and with emphasis on situations in which the crossed products are simple and at least close to the class of C*-algebras expected to be classifiable in the sense of the Elliott program. They are aimed at graduate students who have had a one semester or one year course on the general theory of C*-algebras. (We give more details on the prerequisites later in this section.) These notes are not intended as a reference work. Our emphasis is on explaining ideas and methods, rather than on giving complete proofs. For some results, different proofs are given at different locations in these notes, or special cases are proved of results which are proved later in greater generality by quite different methods. For others, some of the main ideas are explained and simpler versions of some of the relevant lemmas are proved, but we refer to the research papers for the full proofs. Other results and calculations are left as exercises; the reader is strongly encouraged to do many of these, to develop facility with the material. Yet other results, needed for the proofs of the theorems described here but not directly related to dynamics, are quoted with only some general description, or with no background at all.
Thierry Giordano, David Kerr, N. Christopher Phillips, Andrew Toms

Chapter 8. Group Actions

This is the first of three sections devoted to examples of group actions.
In this section, we give examples of actions on commutative C*-algebras. In Section 8.2 we give a variety of examples of actions on noncommutative C*- algebras, and in Section 8.3 we give an additional collection of examples of actions that are similar to gauge actions.
Thierry Giordano, David Kerr, N. Christopher Phillips, Andrew Toms

Chapter 9. Group C*-algebras and Crossed Products

The main focus of these notes is the structure of certain kinds of crossed products. The C*-algebra of a group is a special case of a crossed product—it comes from the trivial action of the group on ℂ—but not one of the ones we are mainly concerned with. We devote this section and Section 9.3 to group C*-algebras anyway, in order to provide an introduction to crossed products in a simpler case, and because understanding the group C*-algebra is helpful, at least at a heuristic level, for understanding more general crossed products. Section 9.4 treats crossed product C*-algebras and Section 9.5 treats reduced crossed product C*-algebras. In Section 9.6 we give a number of explicit computations of crossed product C*- algebras. The brief Section 9.2 contains a proof that the reduced C*-algebra of a finitely generated nonabelian free group is simple.
Thierry Giordano, David Kerr, N. Christopher Phillips, Andrew Toms

Chapter 10. Some Structure Theory for Crossed Products by Finite Groups

Our main interest is in structural results for crossed products. We want simplicity, but we really want much more than that. We particularly want theorems showing that certain crossed products are in classes of C*-algebras known to be covered by the Elliott classification program, so that the crossed product can be identified up to isomorphism by computing its K-theory and other invariants. In many cases, one settles for related weaker structural results, such as stable rank one, real rank zero, order on traces determined by projections, strict comparison of positive elements, or Z-stability. Some results with conclusions of this sort are stated in these notes, but mostly without proof.
Thierry Giordano, David Kerr, N. Christopher Phillips, Andrew Toms

Chapter 11. An Introduction to Crossed Products by Minimal Homeomorphisms

In this section, we discuss free and essentially free minimal actions of countable discrete groups on compact metric spaces, with emphasis on minimal homeomorphisms (actions of ℤ). We give two simplicity proofs, using very different methods. One works for free minimal actions, and the method gives further information, as well as some information when the action is not minimal; see Theorems 11.1.20 and 11.1.22. The second proof is a special case of a more general simplicity theorem; the case we prove allows some simplification of the argument. Our theorem is Theorem 11.1.10, and its proof is given before Theorem 11.1.25. The full theorem is stated as Theorem 11.1.25. Both proofs end with an argument related to the proof that Kishimoto’s condition (see Definition 10.4.20) implies simplicity of the crossed product (see Theorem 10.4.22), but the two proofs use quite different routes to get there.
Thierry Giordano, David Kerr, N. Christopher Phillips, Andrew Toms

Chapter 12. An Introduction to Large Subalgebras and Applications to Crossed Products

In this part, we give an introduction to large subalgebras of C*-algebras and some applications. Much of the text of this part is taken directly from [215], which is a survey of applications of large subalgebras based on lectures given at the University of Wyoming in the summer of 2015. That survey assumes much more background than these notes (it starts with the material here), there are some differences in the organization, and it contains some open problems and other discussion omitted here because they are too far off the topic of these notes.
Thierry Giordano, David Kerr, N. Christopher Phillips, Andrew Toms

C*-Algebras and Topological Dynamics: Finite Approximation and Paradoxicality


Chapter 13. Introduction

The aim of these notes is to survey several recent developments at the crossed product interface of the subjects of C*-algebras and group actions on compact spaces, especially in connection with the classification program for separable nuclear C*-algebras. Groups and group actions have from the beginning provided a rich source of examples in the theory of operator algebras, and the struggle to obtain an algebraic understanding of dynamical phenomena has to a great extent driven, and continues to drive, the development of structure and classification theory for both von Neumann algebras and C*-algebras. While our focus is on the topological realm of C*-algebras, we have nevertheless endeavored to take a broad perspective that incorporates both the measurable and the topological in a unifying framework. This enables us not only to illuminate the conceptual similarities and technical differences between the two sides, but also to emphasize that topological-dynamical and C*-algebraic concepts themselves can range from the more measure-theoretic (like entropy and nuclearity, which involve weak-type approximation of multiplicative structure or norm approximation of linear structure) to the more topological (like periodicity and approximate finite-dimensionality, which involve norm approximation of multiplicative structure). Thinking in such terms can be helpful for predicting and understanding the role of various phenomena in C*-classification theory.
Thierry Giordano, David Kerr, N. Christopher Phillips, Andrew Toms

Chapter 14. Internal Measure-Theoretic Phenomena

The notion of amenability in its most basic combinatorial sense captures the idea of internal finite approximation from a measure-theoretic perspective. It plays a pivotal role not only in combinatorial and geometric group theory but also in the theory of operator algebras through its various linear manifestations like hyperfiniteness, semidiscreteness, injectivity, and nuclearity. In this section we will review the theory of amenability for discrete groups (see [34, 76] for general references), and then move in Section 14.2 to amenable actions and their reduced crossed products.
Thierry Giordano, David Kerr, N. Christopher Phillips, Andrew Toms

Chapter 15. External Measure-Theoretic Phenomena

As discussed in Section 14.1, for discrete groups the basic idea of internal measuretheoretic finite approximation is captured by the Følner set characterization of amenability. At the same time we can view Følner sets as furnishing external finite approximations in the following way. Let F be a nonempty finite subset of a discrete group G.
Thierry Giordano, David Kerr, N. Christopher Phillips, Andrew Toms

Chapter 16. Internal Topological Phenomena

The topological analogue of amenability for discrete groups is local finiteness. In contrast to the setting of C*-algebras, which we will turn to below, for discrete groups the topological notion of perturbation is trivial and thus, unlike the combinatorial measure-theoretic viewpoint, does not give us anything new beyond the merely group-theoretic. The group G is said to be locally finite if every finite subset of G generates a finite subgroup. Equivalently, G is the increasing union of finite subgroups. Obviously every finite group is locally finite. An example of a countably infinite locally finite group is the group of all permutations of N which fix all but finitely many elements. There are uncountably many pairwise nonisomorphic countable locally finite groups, and there is a countable locally finite group U which has the universal property that it contains a copy of every finite group and any two monomorphisms of a finite group into U are conjugate by an inner automorphism [47]. Note that every locally finite group is torsion. The converse is the general Burnside problem and is false, as was shown by Golod. In fact a torsion group need not even be amenable, which is the measure-theoretic analogue of local finiteness to be discussed below.
Thierry Giordano, David Kerr, N. Christopher Phillips, Andrew Toms

Chapter 17. External Topological Phenomena

Complementing the internal finite modelling property of local finiteness is the external property of local embeddability into finite groups, which is the topological (i.e., purely group-theoretic, since our groups are discrete) analogue of soficity. The group G is said to be LEF (locally embeddable into finite groups) if for every finite set FG there is a finite group H and a map σ : GH such that σ(st) = σ(s)σ(t) for all s, tF and σ|F is injective. This notion was introduced by Gordon and Vershik in [33].
Thierry Giordano, David Kerr, N. Christopher Phillips, Andrew Toms

Minimal Topological Systems and Orbit Equivalence


Chapter 18. Introduction

In 1959, Dye [19] introduced the notion of orbit equivalence and proved that any two ergodic finite measure preserving transformations on a Lebesgue space are orbit equivalent. In [20], he had also conjectured that an arbitrary ergodic action of a discrete amenable group is orbit equivalent to a ℤ–action. This conjecture was proved in Ornstein–Weiss [73]. The most general case was proved in Connes– Feldman–Weiss [13] by establishing that an amenable nonsingular countable equivalence relation \(\mathcal{R}\) can be generated by a single transformation, or equivalently, is hyperfinite, i.e., \(\mathcal{R}\) is up to a null set, a countable increasing union of finite equivalence relations.
Thierry Giordano, David Kerr, N. Christopher Phillips, Andrew Toms

Chapter 19. Cantor Dynamics

The main goal of this chapter is the presentation of the Bratteli–Vershik model developed by R. Herman, I.F. Putnam and C.F. Skau in their remarkable paper [46].
In the first section, we recall definitions of dynamical concepts, that will be often used. The proofs of the results of this section can be found in any standard textbook on topological dynamics; see, for example, [59, 85, 86].
In the second section, we will specialize to Cantor minimal systems.
Thierry Giordano, David Kerr, N. Christopher Phillips, Andrew Toms

Chapter 20. Etale Equivalence Relations

In the first section of this chapter, we will first recall the definition and the first properties of étale equivalence relations.We restrict our presentation to the notions we will need in the next chapters; for more details see, for example, [74, 78, 80]. In the second section, we recall the definitions of isomorphism and orbit equivalence of étale equivalence relations. In the second section, we define AF-equivalence relations, a rich but also tractable class of étale equivalence relations that we will classify up to isomorphism and up to orbit equivalence, respectively, in Section 20.5 and Chapter 22.
Thierry Giordano, David Kerr, N. Christopher Phillips, Andrew Toms

Chapter 21. The Absorption Theorem

The absorption theorem is the main result in this chapter and is the key tool we will use to classify up to orbit equivalence minimal AF-equivalence relations and étale equivalence relations associated to minimal actions on the Cantor set of finitely generated abelian groups. This key result shows that a “small” extension of a minimal AF-equivalence relation is orbit equivalent to the original one.
Thierry Giordano, David Kerr, N. Christopher Phillips, Andrew Toms

Chapter 22. Orbit Equivalence of AF-Equivalence Relations

In Definition 20.4.33, we introduced the invariant Dm(X,\(\mathcal{R}\)) for an étale equivalence relation \(\mathcal{R}\) on a totally disconnected, compact Hausdorff space X, and showed in Proposition 20.4.40 that it is an orbit equivalence invariant. In this chapter, we will show that this invariant is complete for the class of AF equivalence relations. The classification up to orbit equivalence of minimal AF-equivalence relations will be a corollary of the following theorem from Putnam [78].
Thierry Giordano, David Kerr, N. Christopher Phillips, Andrew Toms

Chapter 23. Orbit Equivalence of Minimal Actions of a Finitely Generated Abelian Group

In measurable dynamics, the study of orbit equivalence, initiated by Dye [19], was developed by Krieger [56], Ornstein–Weiss [73] and Connes–Feldman–Weiss [13] among many others in the amenable case. The strategy of their proofs consisted of showing that any amenable measurable equivalence relation is orbit equivalent to a hyperfinite measurable equivalence relation and classifying these ones. In the nonsingular case, the complete invariant of orbit equivalence is an ergodic flow, the so-called associated flow; see Krieger [56].
Thierry Giordano, David Kerr, N. Christopher Phillips, Andrew Toms

Chapter 24. Orbit and Strong Orbit Realization for Minimal Homeomorphisms: Ormes’ Results

In the first part of this chapter we will present Jewett–Krieger type realization results of an ergodic dynamical system by a Cantor minimal system in a prescribed orbit equivalence class. Nic Ormes proved them in his thesis and they are published in [71].
Thierry Giordano, David Kerr, N. Christopher Phillips, Andrew Toms

Chapter 25. Full Groups

The notion of full group was introduced in 1959 by H. Dye in his study of orbit equivalence of measured dynamical systems [19] and [20].
In the first part of this chapter we will review the definition of full groups in the measurable case. We will then describe some of their properties. In the second part, we will present the different full groups associated to a topological dynamical system (on the Cantor set) and study their properties in comparison with the measurable case. Since their introduction in [39], the so-called topological full groups have been intensively studied and outstanding results obtained. As remarkable surveys on topological full groups and their properties have recently been written (see de Cornulier [14], Juschenko [53], and Matui [66]), we will review only in this section the properties of full groups.
Thierry Giordano, David Kerr, N. Christopher Phillips, Andrew Toms
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